Best Known (23, 23+33, s)-Nets in Base 64
(23, 23+33, 242)-Net over F64 — Constructive and digital
Digital (23, 56, 242)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 16, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (7, 40, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- digital (0, 16, 65)-net over F64, using
(23, 23+33, 288)-Net in Base 64 — Constructive
(23, 56, 288)-net in base 64, using
- t-expansion [i] based on (22, 56, 288)-net in base 64, using
- 35 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
- base change [i] based on digital (9, 78, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 78, 288)-net over F128, using
- 35 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
(23, 23+33, 343)-Net over F64 — Digital
Digital (23, 56, 343)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6456, 343, F64, 3, 33) (dual of [(343, 3), 973, 34]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(6453, 342, F64, 3, 33) (dual of [(342, 3), 973, 34]-NRT-code), using
- extended algebraic-geometric NRT-code AGe(3;F,992P) [i] based on function field F/F64 with g(F) = 20 and N(F) ≥ 342, using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(6453, 342, F64, 3, 33) (dual of [(342, 3), 973, 34]-NRT-code), using
(23, 23+33, 513)-Net in Base 64
(23, 56, 513)-net in base 64, using
- 4 times m-reduction [i] based on (23, 60, 513)-net in base 64, using
- base change [i] based on digital (8, 45, 513)-net over F256, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- K1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F256 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- base change [i] based on digital (8, 45, 513)-net over F256, using
(23, 23+33, 174551)-Net in Base 64 — Upper bound on s
There is no (23, 56, 174552)-net in base 64, because
- 1 times m-reduction [i] would yield (23, 55, 174552)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 2187 434878 989981 093127 855709 859224 412542 424282 533892 261342 583060 352060 283952 621380 449854 135426 371433 > 6455 [i]